import numpy as np

def loss$($w$,$ b$,$ xTr$,$ yTr$,$ C$)$:

$"""$

INPUT:

w : d dimensional weight vector

b : scalar $($bias$)$

xTr : nxd dimensional matrix $($each row is an input vector$)$

yTr : n dimensional vector $($each entry is a label$)$

C : scalar $($constant that controls the tradeoff between l$2-$regularizer and hinge$-$loss$)$

OUTPUTS:

loss : the total loss obtained with $($w$,$ b$)$ on xTr and yTr $($scalar$)$

$"""$

Tests:

xTr$\_$test, yTr$\_$test $=$ generate$\_$data$()$

n$,$ d $=$ xTr$\_$test.shape

# Check whether your loss$()$ returns a scalar

def loss$\_$test$1()$:

w $=$ np$.$random.rand$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $10)$

return np$.$isscalar$($loss$\_$val$)$

# Check whether your loss$()$ returns a nonnegative scalar

def loss$\_$test$2()$:

w $=$ np$.$random.rand$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $10)$

return loss$\_$val $>=0$

# Check whether you implement l$2-$regularizer correctly

def loss$\_$test$3()$:

w $=$ np$.$random.rand$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $0)$

loss$\_$val$\_$grader $=$ loss$\_$grader$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $0)$

return $($np$.$linalg.norm$($loss$\_$val $-$ loss$\_$val$\_$grader$)<1$e$-5)$

# Check whether you implemented the squared hinge loss and not the standard hinge loss

# Note, loss$\_$grader$\_$wrong is the wrong implementation of the standard hinge$-$loss,

# so the results should NOT match.

def loss$\_$test$4()$:

w $=$ np$.$random.randn$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $1)$

badloss $=$ loss$\_$grader$\_$wrong$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $1)$

return not$($np$.$linalg.norm$($loss$\_$val $-$ badloss$)<1$e$-5)$

# Check whether you implement square hinge loss correctly

def loss$\_$test$5()$:

w $=$ np$.$random.randn$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $10)$

loss$\_$val$\_$grader $=$ loss$\_$grader$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $10)$

return $($np$.$linalg.norm$($loss$\_$val $-$ loss$\_$val$\_$grader$)<1$e$-5)$

# Check whether you implement loss correctly

def loss$\_$test$6()$:

w $=$ np$.$random.randn$($d$)$

b $=$ np$.$random.rand$(1)$

loss$\_$val $=$ loss$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $100)$

loss$\_$val$\_$grader $=$ loss$\_$grader$($w$,$ b$,$ xTr$\_$test, yTr$\_$test, $100)$

return $($np$.$linalg.norm$($loss$\_$val $-$ loss$\_$val$\_$grader$)<1$e$-5)$

runtest$($loss$\_$test$1,$'loss$\_$test$1\text{'})$

runtest$($loss$\_$test$2,$'loss$\_$test$2\text{'})$

runtest$($loss$\_$test$3,$'loss$\_$test$3\text{'})$

runtest$($loss$\_$test$4,$'loss$\_$test$4\text{'})$

runtest$($loss$\_$test$5,$'loss$\_$test$5\text{'})$

runtest$($loss$\_$test$6,$'loss$\_$test$6\text{'})$